Radical of Unit Ideal

Theorem

Let $A$ be a commutative ring with unity.

Let $\ideal 1$ be its unit ideal.

Then its radical equals $\ideal 1$:

$\map \Rad {\ideal 1} = \ideal 1$.

Proof

By definition of ideal:

$\map \Rad {\ideal 1} \subseteq A$

By Ideal of Ring is Contained in Radical:

$\ideal 1 = A \subseteq \Rad {\ideal 1}$.

By definition of set equality:

$\map \Rad {\ideal 1} = \ideal 1$

$\blacksquare$